Statistical Model Based On Matrix Decomposition And Tensor Decomposition And Its Application

With the advancement of modern science and technology and the development of digitization,big data has gradually become popular.These data are not only large in scale,but also complex in structure.They are generally high-dimensional and usually can be expressed in the form of tensors.The traditional method of processing this kind of data is to express it as a large-scale matrix through dimensionality reduction or matrixization,and then realize data analysis and data mining through methods such as matrix decomposition.However,doing so will not only destroy the spatial structure of the data,but also cause the dimensional disaster of the data.In order to solve this problem,various data analysis methods have been extensively studied.Among them,tensor decomposition has attracted people's attention due to its unique advantages.This thesis will use tensor decomposition to study the following two parts:In the first part of the thesis we study the tensor response regression model and the least square estimation of its coefficient tensor.In order to improve the estimation accuracy of the coefficient tensor of the model,two new tensor response regression models are constructed by CP decomposition and Tucker decomposition.These two models can not only capture the internal spatial structure information of tensor data,but also greatly reduce the number of parameters to be estimated.Then,the parameter estimation algorithm corresponding to the model is given.Finally,Monte Carlo numerical experiments show that the estimation accuracy of the coefficient tensors of the two improved regression models is significantly improved,and the estimation accuracy of the coefficient tensors of the tensor response regression model based on Tucker decomposition is the best.In the second part of the thesis we study the application of nonnegative tensor factorization in face recognition.In order to further improve the accuracy of face recognition,a face recognition algorithm based on orthogonal sparse constrained nonnegative tensor decomposition is proposed.Firstly,orthogonal sparsity constraint is added to the traditional nonnegative tensor factorization to reduce the correlation between base images and obtain sparse coding.Secondly,the original face image and the decomposed base image are used to calculate the low dimensional feature representation of the face.Finally,cosine similarity is used to measure the similarity between low dimensional features to judge whether two face images represent the same person.Through experiments in AR database and ORL database,it is found that the improved algorithm can achieve better recognition effect.